PUMaC 2016 · 组合（A 组） · 第 7 题

PUMaC 2016 — Combinatorics (Division A) — Problem 7

The Dinky is a train connecting Princeton to the outside world. It runs on an odd schedule: the train arrives once every one-hour block at some uniformly random time (once at a random time between 9am and 10am, once at a random time between 10am and 11am, and so on). One day, Emilia arrives at the station, at some uniformly random time, and does not know the time. She expects to wait for y minutes for the next train to arrive. After waiting for an hour, a train has still not come. She now expects to wait for z minutes. Find yz .

解析

Say that Emilia arrives at the station at hour x , 0 < x < 1. (for example, if she arrives at 1 8:30, then x = ). Then, the probability that she misses both trains that could possibly arrive 2 in the next hour is x (1 − x ). The expected amount of time she then waits for the second train 1 is (1 − x ). 2 We can picture the weighted amount of time (over probability) as the tetrahedra (0 , 0 , 0), ( ) ( ) 1 1 (1 , 0 , 0), 0 , 0 , , 0 , 1 , , where the value of z is the expected amount of time in hours that 2 2 1 she waits if she arrives at time x . The average value of z is then ; translating z to minutes, 4 we get z = 15. Computing y similarly, we find y = 35. Thus, the answer is 35 · 15 = 525 . Problem written by Bill Huang and Zhuo Qun Song.